Bergman kernel

Let Gn be a domain ( And let A2(G) be the Bergman space. For a fixed zG, the functional ff(z) is a boundedPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath linear functionalMathworldPlanetmath. By the Riesz representation theorem (as A2(G) is a Hilbert spaceMathworldPlanetmath) there exists an element of A2(G) that represents it, and let’s call that element kzA2(G). That is we have that f(z)=f,kz. So we can define the Bergman kernelMathworldPlanetmath.


The function


is called the Bergman kernel.

By definition of the inner productMathworldPlanetmath in A2(G) we then have that for fA2(G)


where dV is the volume measure.

As the A2(G) space is a subspaceMathworldPlanetmathPlanetmath of L2(G,dV) which is a separable Hilbert space then A2(G) also has a countable orthonormal basisMathworldPlanetmath, say {φj}j=1.


We can compute the Bergman kernel as


where the sum converges uniformly on compact subsets of G×G.

Note that integration against the Bergman kernel is just the orthogonal projection from L2(G,dV) to A2(G). So not only is this kernel reproducing for holomorphic functionsMathworldPlanetmath, but it will produce a holomorphic function when we just feed in any L2(G,dV) function.


  • 1 Steven G. Krantz. , AMS Chelsea Publishing, Providence, Rhode Island, 1992.
Title Bergman kernel
Canonical name BergmanKernel
Date of creation 2013-03-22 15:04:45
Last modified on 2013-03-22 15:04:45
Owner jirka (4157)
Last modified by jirka (4157)
Numerical id 7
Author jirka (4157)
Entry type Definition
Classification msc 32A25
Related topic BergmanSpace
Related topic BergmanMetric